I use the same type lamp and everyone I have gotten lets off that burning smell at first. It should stop after the lamp has been on for about 24 hrs.
If the smell bothers you go put it outside or another room and let it run, it should go away if not take it back and return it.
As for the heating pad as long as the temps don't increase it is ok to leave on 24/7 as most of us do.
I was a little freaked at first about leaving things running also but just make sure you use proper cords and don't overload outlets and everything should be fine.